CONDUCTOR INEQUALITIES AND CRITERIA FOR
SOBOLEV-LORENTZ TWO-WEIGHT INEQUALITIES
ŞERBAN COSTEA AND VLADIMIR MAZ’YA
In memory of S. L. Sobolev
Abstract. In this paper we present integral conductor inequalities connecting the
Lorentz p, q-(quasi)norm of a gradient of a function to a one-dimensional integral of
the p, q-capacitance of the conductor between two level surfaces of the same function.
These inequalities generalize an inequality obtained by the second author in the case
of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient
conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.
1. Introduction
During the last decades Sobolev-Lorentz function spaces, which include classical
Sobolev spaces, attracted attention not only as an interesting mathematical object,
but also as a tool for a finer tuning of properties of solutions to partial differential
equations. (See [Alb], [AFT1], [AFT2], [BBGGPV], [Cia], [CP], [Cos], [DHM], [HL],
[KKM], [ST], et al.)
In the present paper we generalize the inequality
Z
Z ∞
p
(1)
capp (Mat , Mt )d(t ) ≤ c(a, p) |∇f |p dx
Ω
0
to Sobolev-Lorentz spaces. Here f ∈ Lip0 (Ω), i.e. f is an arbitrary Lipschitz function
compactly supported in the open set Ω ⊂ Rn , while Mt is the set {x ∈ Ω : |f (x)| > t}
with t > 0. Inequality (1) was obtained in [M1]. (See also [M3, Chapter 2].) It has
various extensions and applications to the theory of Sobolev-type spaces on domains
in Rn , Riemannian manifolds, metric and topological spaces, to linear and nonlinear
partial differential equations, Dirichlet forms, and Markov processes etc. (See [Ad],
[AH], [AP], [AX1], [AX2], [Ai], [CS], [Dah], [DKX], [Fi], [FU1], [FU2], [Gr], [Haj],
[Han], [HMV], [Ka], [Ko1], [Ko2], [Mal], [M1], [M2], [M4], [M5], [MN], [MP], [Ne], [Ra],
[Ta], [V1], [V2], [Vo], et al). In the sequel, we prove the inequalities
Z ∞
(2)
cap(Mat , Mt )d(tp ) ≤ c(a, p, q)||∇f ||pLp,q (Ω,mn ;Rn ) when 1 ≤ q ≤ p
0
and
Z
(3)
0
∞
capp,q (Mat , Mt )q/p d(tq ) ≤ c(a, p, q)||∇f ||qLp,q (Ω,mn ;Rn ) when p < q < ∞
for all f ∈ Lip0 (Ω).
2000 Mathematics Subject Classification. Primary: 31C15, 46E35.
Key words and phrases. Sobolev-Lorentz spaces, conductor capacitance, conductor inequalities,
two-weight integral inequalities.
1
The proof of (2) and (3) is based on the superadditivity of the p, q-capacitance, also
justified in this paper.
From (2) and (3) we derive necessary and sufficient conditions for certain two-weight
inequalities involving Sobolev-Lorentz norms, generalizing results obtained in [M4] and
[M5]. Specifically, let µ and ν be two locally finite nonnegative measures on Ω and
let p, q, r, s be real numbers such that 1 < s ≤ max(p, q) ≤ r < ∞ and q ≥ 1. We
characterize the inequality
(4)
||f ||Lr,max(p,q) (Ω,µ) ≤ A ||∇f ||Lp,q (Ω,mn ;Rn ) + ||f ||Ls,max(p,q) (Ω,ν)
restricted to functions f ∈ Lip0 (Ω) by requiring the condition
(5)
µ(g)1/r ≤ K(capp,q (g, G)1/p + ν(G)1/s )
to be valid for all open bounded sets g and G subject to g ⊂ G, G ⊂ Ω. When n = 1
inequality (4) becomes
||f ||Lr,max(p,q) (Ω,µ) ≤ A ||f 0 ||Lp,q (Ω,m1 ) + ||f ||Ls,max(p,q) (Ω,ν) .
(6)
The requirement that (6) be valid for all functions f ∈ Lip0 (Ω) when n = 1 is shown
to be equivalent to the condition
(7)
µ(σd (x))1/r ≤ K(τ (1−p)/p + ν(σd+τ (x))1/s )
whenever x, d and τ are such that σd+τ (x) ⊂ Ω. Here and throughout the paper σd (x)
denotes the open interval (x − d, x + d) for every d > 0.
2. Preliminaries
Let us introduce some notation, to be used in the sequel. By Ω we denote a nonempty
open subset of Rn , whereas mn stands for the Lebesgue n-measure in Rn , where n ≥ 1
is integer. For a Lebesgue measurable u : Ω → R, supp u is the smallest closed set
such that u vanishes outside supp u. We also define
Lip(Ω) = {ϕ : Ω → R : ϕ is Lipschitz}
Lip0 (Ω) = {ϕ : Ω → R : ϕ is Lipschitz and with compact support in Ω}.
If ϕ ∈ Lip(Ω), we write ∇ϕ for the gradient of ϕ. This notation makes sense, since by
Rademacher’s theorem ([Fed, Theorem 3.1.6]) every Lipschitz function on Ω is mn -a.e.
differentiable.
Throughout this section we will assume that m ≥ 1 is a positive integer and that
(Ω, µ) is a measure space. Let f : Ω → Rn be a µ-measurable function. We define µ[f ] ,
the distribution function of f as follows (see [BS, Definition II.1.1]):
µ[f ] (t) = µ({x ∈ Ω : |f (x)| > t}),
t ≥ 0.
We define f ∗ , the nonincreasing rearrangement of f by
f ∗ (t) = inf{v : µ[f ] (v) ≤ t},
t ≥ 0.
(See [BS, Definition II.1.5].) We notice that f and f ∗ have the same distribution
function. Moreover, for every positive α we have
(|f |α )∗ = (|f |∗ )α
2
and if |g| ≤ |f | a.e. on Ω, then g ∗ ≤ f ∗ . (See [BS, Proposition II.1.7].) We also define
f ∗∗ , the maximal function of f ∗ by
Z
1 t ∗
∗∗
f (t) = mf ∗ (t) =
f (s)ds, t > 0.
t 0
(See [BS, Definition II.3.1].)
Throughout this paper, we will denote the Hölder conjugate of p ∈ [1, ∞] by p0 .
The Lorentz space Lp,q (Ω, µ; Rn ), 1 < p < ∞, 1 ≤ q ≤ ∞, is defined as follows:
Lp,q (Ω, µ; Rn ) = {f : Ω → Rn : f is µ-measurable and ||f ||Lp,q (Ω,µ;Rn ) < ∞},
where
||f ||Lp,q (Ω,µ;Rn ) = || |f | ||p,q =
 Z




∞
dt
f (t))
t
1/p ∗
(t
0
q
1/q



 sup tµ[f ] (t)1/p = sup s1/p f ∗ (s)
t>0
1≤q<∞
q = ∞.
s>0
(See [BS, Definition IV.4.1].) We omit Rn in the notation of function spaces for the
scalar case, i.e. for n = 1.
If 1 ≤ q ≤ p, then || · ||Lp,q (Ω,µ;Rn ) represents a norm, but for p < q ≤ ∞ it represents
a quasinorm, equivalent to the norm || · ||L(p,q) (Ω,µ;Rn ) , where
 Z ∞
1/q

dt
1/p
∗∗
q


(t f (t))
1≤q<∞

t
0
||f ||L(p,q) (Ω,µ;Rn ) = || |f | ||(p,q) =



 sup t1/p f ∗∗ (t)
q = ∞.
t>0
(See [BS, Definition IV.4.4].) Namely, from [BS, Lemma IV.4.5] we have that
|| |f | ||Lp,q (Ω,µ) ≤ || |f | ||L(p,q) (Ω,µ) ≤ p0 || |f | ||Lp,q (Ω,µ)
for every q ∈ [1, ∞] and every µ-measurable function f : Ω → Rn .
It is known that (Lp,q (Ω, µ; Rn ), || · ||Lp,q (Ω,µ;Rn ) ) is a Banach space for 1 ≤ q ≤ p,
while (Lp,q (Ω, µ; Rn ), || · ||L(p,q) (Ω,µ;Rn ) ) is a Banach space for 1 < p < ∞, 1 ≤ q ≤ ∞.
Remark 2.1. It is also known (see [BS, Proposition IV.4.2]) that for every p ∈ (1, ∞)
and 1 ≤ r < s ≤ ∞ there exists a constant C(p, r, s) such that
(8)
|| |f | ||Lp,s (Ω,µ) ≤ C(p, r, s)|| |f | ||Lp,r (Ω,µ)
for all measurable functions f ∈ Lp,r (Ω, µ; Rn ) and all integers n ≥ 1. In particular,
the embedding Lp,r (Ω, µ; Rn ) ,→ Lp,s (Ω, µ; Rn ) holds.
2.1. The subadditivity and superadditivity of the Lorentz quasinorms. In
the second part of this paper, we will prove a few results by relying on the superadditivity of the Lorentz p, q-quasinorm. Therefore we recall the known results and
present new results concerning the superadditivity and the subadditivity of the Lorentz
p, q-quasinorm.
The superadditivity of the Lorentz p, q-norm in the case 1 ≤ q ≤ p was stated in
[CHK, Lemma 2.5].
3
Proposition 2.2. (See [CHK, Lemma 2.5].) Let (Ω, µ) be a measure space. Suppose
1 ≤ q ≤ p. Let {Ei }i≥1 be a collection of pairwise disjoint measurable subsets of Ω with
E0 = ∪i≥1 Ei and let f ∈ Lp,q (Ω, µ). Then
X
||χEi f ||pLp,q (Ω,µ) ≤ ||χE0 f ||pLp,q (Ω,µ) .
i≥1
We obtain a similar result concerning the superadditivity in the case 1 < p < q < ∞.
Proposition 2.3. Let (Ω, µ) be a measure space. Suppose 1 < p < q < ∞. Let {Ei }i≥1
be a collection of pairwise disjoint measurable subsets of Ω with E0 = ∪i≥1 Ei and let
f ∈ Lp,q (Ω, µ). Then
X
||χEi f ||qLp,q (Ω,µ) ≤ ||χE0 f ||qLp,q (Ω,µ) .
i≥1
Proof. For every i = 0, 1, 2, . . . we let fi = χEi f, where χEi is the characteristic function
of Ei . We can assume without loss of generality that all the functions fi are nonnegative.
We have (see [KKM, Proposition 2.1])
Z ∞
q
sq−1 µ[fi ] (s)q/p ds,
||fi ||Lp,q (Ω,µ) = p
0
where µ[fi ] is the distribution function of fi , i = 0, 1, 2, . . . . From the definition of f0
we obviously have
X
(9)
µ[f0 ] (s) =
µ[fi ] (s) for every s > 0
i≥1
which implies, since 1 < p < q < ∞, that
X
µ[f0 ] (s)q/p ≥
µ[fi ] (s)q/p for every s > 0.
i≥1
This yields
||f0 ||qLp,q (Ω,µ)
Z
= p
∞
s
q−1
q/p
µ[f0 ] (s)
0
ds ≥ p
X
sq−1 (
µ[fi ] (s)q/p )ds
0
X Z
=
p
i≥1
∞
Z
∞
sq−1 µ[fi ] (s)q/p ds =
0
i≥1
X
||f0 ||qLp,q (Ω,µ) .
i≥1
This finishes the proof of the superadditivity in the case 1 < p < q < ∞.
We have a similar result for the subadditivity of the Lorentz p, q-quasinorm. When
1 < p < q ≤ ∞ we obtain a result that generalizes [Cos, Theorem 2.5].
Proposition 2.4. Let (Ω, µ) be a measure space. Suppose 1 < p < q ≤ ∞. Let {Ei }i≥1
be a collection of pairwise disjoint measurable subsets of Ω with E0 = ∪i≥1 Ei and let
f ∈ Lp,q (Ω, µ). Then
X
||χEi f ||pLp,q (Ω,µ) ≥ ||χE0 f ||pLp,q (Ω,µ) .
i≥1
4
Proof. Without loss of generality we can assume that all the functions fi are nonnegative. We have to consider two cases, depending on whether p < q < ∞ or q = ∞.
Suppose p < q < ∞. We have (see [KKM, Proposition 2.1])
p/q
Z ∞
q/p
p
q−1
s µ[fi ] (s) ds
,
||fi ||Lp,q (Ω,µ) = p
0
where µ[fi ] is the distribution function of fi for i = 0, 1, 2, . . . . From (9) we obtain
p/q X Z ∞
p/q
Z ∞
q/p
p
q−1
q/p
q−1
s µ[f0 ] (s) ds
≤
p
s µ[fi ] (s) ds
||f0 ||Lp,q (Ω,µ) =
p
0
=
X
i≥1
0
||fi ||pLp,q (Ω,µ) .
i≥1
Suppose now q = ∞. From (9) we obtain
X
sp µ[f0 ] (s) =
(sp µ[fi ] (s))
i≥1
for every s ≥ 0, which implies
(10)
sp µ[f0 ] (s) ≤
X
||fi ||pLp,∞ (Ω,µ)
i≥1
for every s ≥ 0. By taking the supremum over all s ≥ 0 in (10), we get the desired
conclusion. This finishes the proof.
3. Sobolev-Lorentz p, q-capacitance
Suppose 1 < p < ∞ and 1 ≤ q ≤ ∞. Let Ω ⊂ Rn be an open set, n ≥ 1. Let K ⊂ Ω
be compact. The Sobolev-Lorentz p, q-capacitance of the conductor (K, Ω) is denoted
by
capp,q (K, Ω) = inf {||∇u||pLp,q (Ω,mn ;Rn ) : u ∈ W (K, Ω)},
where
W (K, Ω) = {u ∈ Lip0 (Ω) : u ≥ 1 in a neighborhood of K}.
We call W (K, Ω) the set of admissible functions for the conductor (K, Ω).
Since W (K, Ω) is closed under truncations from below by 0 and from above by 1
and since these truncations do not increase the p, q-quasinorm whenever 1 < p < ∞
and 1 ≤ q ≤ ∞, it follows that we can choose only functions u ∈ W (K, Ω) that satisfy
0 ≤ u ≤ 1 when computing the p, q-capacitance of the conductor (K, Ω).
Lemma 3.1. If Ω is bounded, then we get the same p, q-capacitance for the conductor
(K, Ω) if we restrict ourselves to a bigger set, namely
W1 (K, Ω) = {u ∈ Lip(Ω) ∩ C(Ω) : u ≥ 1 on K and u = 0 on ∂Ω}.
Proof. Let u ∈ W1 (K, Ω). We can assume without loss of generality that 0 ≤ u ≤ 1.
e be
Moreover, we can also assume that u = 1 on an open neighborhood U of K. Let U
e ⊂⊂ U. We choose a cutoff Lipschitz function
an open neighborhood of K such that U
e of K. We notice
η, 0 ≤ η ≤ 1 such that η = 1 in Ω \ U and η = 0 in a neighborhood U
5
that 1 − η(1 − u) = u. We notice that there exists a sequence of functions ϕj ∈ Lip0 (Ω)
such that
lim (||ϕj − u||Lp+1 (Ω,mn ) + ||∇ϕj − ∇u||Lp+1 (Ω,mn ;Rn ) ) = 0.
j→∞
Without loss of generality the sequence ϕj can be chosen such that ϕj → u and
∇ϕj → ∇u pointwise a.e. in Ω. Then ψj = 1 − η(1 − ϕj ) is a sequence belonging to
W (K, Ω) and
lim (||ψj − u||Lp+1 (Ω,mn ) + ||∇ψj − ∇u||Lp+1 (Ω,mn ;Rn ) ) = 0.
j→∞
This, Hölder’s inequality for Lorentz spaces, and the behaviour of the Lorentz p, qquasinorm in q yields
lim (||ψj − u||Lp,q (Ω,mn ) + ||∇ψj − ∇u||Lp,q (Ω,mn ;Rn ) ) = 0.
j→∞
The desired conclusion follows.
3.1. Basic properties of the p, q-capacitance. Usually, a capacitance is a monotone
and subadditive set function. The following theorem will show, among other things,
that this is true in the case of the p, q-capacitance. We follow [Cos] for (i)-(vi). In
addition we will prove some superadditivity properties of the p, q-capacitance.
Theorem 3.2. Suppose 1 < p < ∞ and 1 ≤ q ≤ ∞. Let Ω ⊂ Rn be open. The set
function K 7→ capp,q (K, Ω), K ⊂ Ω, K compact, enjoys the following properties:
(i) If K1 ⊂ K2 , then capp,q (K1 , Ω) ≤ capp,q (K2 , Ω).
(ii) If Ω1 ⊂ Ω2 are open and K is a compact subset of Ω1 , then
capp,q (K, Ω2 ) ≤ capp,q (K, Ω1 ).
(iii) If Ki is a decreasing sequence of compact subsets of Ω with K =
T∞
i=1
Ki , then
capp,q (K, Ω) = lim capp,q (Ki , Ω).
i→∞
S
(iv) If Ωi is an increasing sequence of open sets with ∞
i=1 Ωi = Ω and K is a compact
subset of Ω1 , then
capp,q (K, Ω) = lim capp,q (K, Ωi ).
i→∞
Sk
(v) Suppose p ≤ q ≤ ∞. If K = i=1 Ki ⊂ Ω then
capp,q (K, Ω) ≤
k
X
capp,q (Ki , Ω),
i=1
where k ≥ 1 is a positive integer. S
(vi) Suppose 1 ≤ q < p. If K = ki=1 Ki ⊂ Ω then
q/p
capp,q (K, Ω)
≤
k
X
capp,q (Ki , Ω)q/p ,
i=1
where k ≥ 1 is a positive integer.
(vii) Suppose 1 ≤ q ≤ p. Suppose Ωi , . . . , Ωk are k pairwise disjoint open sets and
Ki are compact subsets of Ωi for i = 1, . . . , k. Then
capp,q (∪ki=1 Ki , ∪ki=1 Ωi )
≥
k
X
i=1
6
capp,q (Ki , Ωi ).
(viii) Suppose p < q < ∞. Suppose Ωi , . . . , Ωk are k pairwise disjoint open sets and
Ki are compact subsets of Ωi for i = 1, . . . , k. Then
capp,q (∪ki=1 Ki , ∪ki=1 Ωi )q/p
≥
k
X
capp,q (Ki , Ωi )q/p .
i=1
(ix) Suppose 1 ≤ q < ∞. If Ω1 and Ω2 are two disjoint open sets and K ⊂ Ω1 , then
capp,q (K, Ω1 ∪ Ω2 ) = capp,q (K, Ω1 ).
Proof. Properties (i)-(vi) are proved by duplicating the proof of [Cos, Theorem 3.2], so
we will prove only (vii)-(ix).
In order to prove (vii) and (viii), it is enough to assume that k = 2. A finite induction
on k would prove each of these claims. So we assume that k = 2. Let u ∈ Lip0 (Ω1 ∪ Ω2 )
and let ui = χΩi u, i = 1, 2. We let vi be the restriction of u to Ωi for i = 1, 2. Then
vi ∈ Lip0 (Ωi ) for i = 1, 2. We notice that ui can be regarded as the extension of vi
by 0 to Ω1 ∪ Ω2 for i = 1, 2. We notice that u ∈ W (K1 ∪ K2 , Ω1 ∪ Ω2 ) if and only if
vi ∈ W (Ki , Ωi ) for i = 1, 2.
Suppose first that 1 ≤ q ≤ p. Since Ω1 and Ω2 are disjoint and u = u1 + u2 with the
functions ui supported in Ωi for i = 1, 2, we obtain via Proposition 2.2
||∇u||pLp,q (Ω1 ∪Ω2 ,mn ;Rn ) ≥ ||∇u1 ||pLp,q (Ω1 ∪Ω2 ,mn ;Rn ) + ||∇u2 ||pLp,q (Ω1 ∪Ω2 ,mn ;Rn )
= ||∇v1 ||pLp,q (Ω1 ,mn ;Rn ) + ||∇v2 ||pLp,q (Ω2 ,mn ;Rn ) .
This proves (vii).
Suppose now that p < q < ∞. Since Ω1 and Ω2 are disjoint and u = u1 + u2 with
the functions ui supported in Ωi for i = 1, 2, we obtain via Proposition 2.3
||∇u||qLp,q (Ω1 ∪Ω2 ,mn ;Rn ) ≥ ||∇u1 ||qLp,q (Ω1 ∪Ω2 ,mn ;Rn ) + ||∇u2 ||qLp,q (Ω1 ∪Ω2 ,mn ;Rn )
= ||∇v1 ||qLp,q (Ω1 ,mn ;Rn ) + ||∇v2 ||qLp,q (Ω2 ,mn ;Rn ) .
This proves (viii).
We see that (ix) follows from (vii) and (ii) when 1 ≤ q ≤ p. (We use (vii) with k = 2
by taking K1 = K and K2 = ∅.) When p < q < ∞, (ix) follows from (viii) and (ii).
(We use (viii) with k = 2 by taking K1 = K and K2 = ∅.) This finishes the proof of
the theorem.
Remark 3.3. The definition of the p, q-capacitance implies
capp,q (K, Ω) = capp,q (∂K, Ω)
whenever K is a compact set in Ω. Moreover, if n = 1 and Ω is an open interval of R,
then
capp,q (K, Ω) = capp,q (H, Ω),
where H is the smallest compact interval containing K.
7
4. Conductor inequalities
Lemma 4.1. Suppose Ω ⊂ Rn is open. Let f ∈ Lip0 (Ω) and let a > 1 be a constant.
For t > 0 we denote Mt = {x ∈ Ω : |f (x)| > t}. Then the function t 7→ capp,q (Mat , Mt )
is upper semicontinuous.
Proof. Let t0 > 0 and ε > 0. Let u ∈ W (Mat , Mt ) be chosen such that
||∇u||pLp,q (Ω,mn ;Rn ) < capp,q (Mat , Mt ) + ε.
Let g be an open neighborhood of Mat such that u ≥ 1 on g. Since g contains the
compact set Mat , there exists δ1 > 0 small such that g ⊃ Ma(t0 −δ1 ) . Let G be an open
set such that supp u ⊂ G ⊂⊂ Mt . There exists a small δ2 > 0 such that G ⊂ Mt0 +δ2 .
Thus we have Ma(t0 −δ) ⊂ g and G ⊂ Mt0 +δ for every δ ∈ (0, min{δ1 , δ2 }). From the
choice of g and G we have that u ∈ W (K, Ω) whenever K ⊂ g and G ⊂ Ω. This and
the choice of u imply that
capp,q (Ma(t0 −δ) , Mt0 +δ ) ≤ capp,q (Mat0 , Mt0 ) + ε
for every δ ∈ (0, min{δ1 , δ2 }). Using the monotonicity of capp,q again, we deduce that
capp,q (Mat , Mt ) ≤ capp,q (Mat0 , Mt0 ) + ε
for every t sufficiently close to t0 . The result follows.
Theorem 4.2. Let Φ denote an increasing convex (not necessarily strictly convex)
function given on [0, ∞), Φ(0) = 0. Suppose a > 1 is a constant.
(i) If 1 ≤ q ≤ p, then
Z ∞
dt
−1
p
Φ
Φ(t capp,q (Mat , Mt ))
≤ c(a, p, q)||∇ϕ||pLp,q (Ω,mn ;Rn )
t
0
for every ϕ ∈ Lip0 (Ω).
(ii) If p < q < ∞, then
Z ∞
q
q/p dt
−1
Φ(t capp,q (Mat , Mt ) )
Φ
≤ c(a, p, q)||∇ϕ||qLp,q (Ω,mn ;Rn )
t
0
for every ϕ ∈ Lip0 (Ω).
Proof. The proof follows [M4]. When p = q we are in the case of the p-capacitance and
for that case the result was proved in [M4, Theorem 1]. So we can assume without loss
of generality that p 6= q. Let ϕ ∈ Lip0 (Ω). We set
Λt (ϕ) =
1
min{(|ϕ| − t)+ , (a − 1)t}.
(a − 1)t
From Lemma 3.1 we notice that
(11)
Λt (ϕ) ∈ W1 (Mat , Mt ) and |∇Λt (ϕ)| =
1
χM \M |∇ϕ| mn -a.e.
(a − 1)t t at
The proof splits now, depending on whether 1 ≤ q < p or p < q < ∞.
We assume first that 1 ≤ q < p. From (11) we have
tp capp,q (Mat , Mt ) ≤
1
||χMt \Mat ∇ϕ||pLp,q (Ω,mn ;Rn ) .
p
(a − 1)
8
Hence
Z
0
∞
dt
Φ(t capp,q (Mat , Mt )) ≤
t
p
Z
∞
Φ(
0
1
dt
||χMt \Mat ∇ϕ||pLp,q (Ω,mn ;Rn ) ) .
p
(a − 1)
t
Let γ denote a locally integrable function on (0, ∞) such that there exist the limits
γ(0) and γ(∞). Then the identity
Z ∞
dt
(γ(t) − γ(at)) = (γ(0) − γ(∞)) log a
(12)
t
0
holds.
We set
1
γ(t) = Φ(
||χMt ∇ϕ||pLp,q (Ω,mn ;Rn ) ).
(a − 1)p
Using the convexity of Φ and Proposition 2.2, we see that
1
||χMt \Mat ∇ϕ||pLp,q (Ω,mn ;Rn ) ) ≤ Φ(γ(t)) − Φ(γ(at)).
Φ(
(a − 1)p
Since
1
γ(0) = Φ(
||∇ϕ||pLp,q (Ω,mn ;Rn ) ) and γ(∞) = 0,
(a − 1)p
we get
Z ∞
dt
1
||∇ϕ||pLp,q (Ω,mn ;Rn ) ).
Φ(tp capp,q (Mat , Mt )) ≤ log a · Φ(
p
t
(a
−
1)
0
This finishes the proof of the case 1 ≤ q < p.
We assume now that p < q < ∞. From (11) we have
1
tq capp,q (Mat , Mt )q/p ≤
||χMt \Mat ∇ϕ||qLp,q (Ω,mn ;Rn ) .
q
(a − 1)
Hence
Z ∞
Z ∞
1
dt
q
q/p dt
Φ(t capp,q (Mat , Mt ) ) ≤
Φ(
||χMt \Mat ∇ϕ||qLp,q (Ω,mn ;Rn ) ) .
q
t
(a − 1)
t
0
0
As before, we let γ denote a locally integrable function on (0, ∞) such that there exist
the limits γ(0) and γ(∞). We set
1
γ(t) = Φ(
||χMt ∇ϕ||qLp,q (Ω,mn ;Rn ) ).
q
(a − 1)
Using the convexity of Φ and Proposition 2.3, we see that
1
Φ(
||χMt \Mat ∇ϕ||qLp,q (Ω,mn ;Rn ) ) ≤ Φ(γ(t)) − Φ(γ(at)).
(a − 1)q
Since
1
||∇ϕ||qLp,q (Ω,mn ;Rn ) ) and γ(∞) = 0,
γ(0) = Φ(
(a − 1)q
we get
Z ∞
dt
1
Φ(tq capp,q (Mat , Mt )q/p ) ≤ log a · Φ(
||∇ϕ||qLp,q (Ω,mn ;Rn ) ).
q
t
(a
−
1)
0
This finishes the proof of the case p < q < ∞. The theorem is proved.
9
Choosing Φ(t) = t, we arrive at the inequalities mentioned in the beginning of this
paper.
Corollary 4.3. Suppose 1 < p < ∞ and 1 ≤ q < ∞. Let a > 1 be a constant. Then
(2) and (3) hold for every ϕ ∈ Lip0 (Ω).
5. Necessary and sufficient conditions for two-weight embeddings
We derive now necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two measures, generalizing results obtained in [M4] and [M5].
Theorem 5.1. Let p, q, r, s be chosen such that 1 < p < ∞, 1 ≤ q < ∞ and 1 < s ≤
max(p, q) ≤ r < ∞. Let Ω be an open set in Rn and let µ and ν be two nonnegative
locally finite measures on Ω.
(i) Suppose that 1 ≤ q ≤ p. The inequality
(13)
||f ||Lr,p (Ω,µ) ≤ A ||∇f ||Lp,q (Ω,mn ;Rn ) + ||f ||Ls,p (Ω,ν)
holds for every f ∈ Lip0 (Ω) if and only if there exists a constant K > 0 such that the
inequality (5) is valid for all open bounded sets g and G that are subject to g ⊂ G ⊂
G ⊂ Ω.
(ii) Suppose that p < q < ∞. The inequality
(14)
||f ||Lr,q (Ω,µ) ≤ A ||∇f ||Lp,q (Ω,mn ;Rn ) + ||f ||Ls,q (Ω,ν)
holds for every f ∈ Lip0 (Ω) if and only if there exists a constant K > 0 such that the
inequality (5) is valid for all open bounded sets g and G that are subject to g ⊂ G ⊂
G ⊂ Ω.
Proof. We suppose first that 1 ≤ q ≤ p. The case q = p was studied in [M5]. Without
loss of generality we can assume that q < p. We choose some bounded open sets g and
G such that g ⊂ G ⊂ G ⊂ Ω and f ∈ W (g, G) with 0 ≤ f ≤ 1. We have
µ(g) ≤ C(r, p) ||f ||rLr,p (Ω,µ)
and
||f ||sLs,p (Ω,ν) ≤ C(s, p) ν(G)
for every f ∈ W (g, G) with 0 ≤ f ≤ 1. The necessity for 1 ≤ q < p is obtained by
taking the infimum over all such functions f that are admissible for the conductor
(g, G).
We prove the sufficiency now when 1 ≤ q < p. Let a ∈ (1, ∞). We have
Z ∞
Z ∞
p
p/r
p
p
p/s
p
a
µ(Mat ) d(t ) ≤ a K1
(capp,q (Mat , Mt ) + ν(Mt ) )d(t ) .
0
0
This and (2) yield the sufficiency for the case 1 ≤ q < p.
Suppose now that p < q < ∞. We choose some bounded open sets g and G such
that g ⊂ G ⊂ G ⊂ Ω and f ∈ W (g, G) with 0 ≤ f ≤ 1. We have
µ(g) ≤ C(r, q) ||f ||rLr,q (Ω,µ)
and
||f ||sLs,q (Ω,ν) ≤ C(s, q) ν(G)
10
for every f ∈ W (g, G) with 0 ≤ f ≤ 1. The necessity for p < q < ∞ is obtained
by taking the infimum over all such functions f that are admissible for the conductor
(g, G).
We prove the sufficiency now when p < q < ∞. Let a ∈ (1, ∞). We have
Z ∞
Z ∞
q/r
q
q
q
q/p
q/s
q
µ(Mat ) d(t ) ≤ a K2
(capp,q (Mat , Mt ) + ν(Mt ) )d(t ) .
a
0
0
This and (3) yield the sufficiency for the case p < q < ∞. The proof is finished.
We look for a simplified necessary and sufficient two-weight imbedding condition
when n = 1. Before we state and prove such a condition for the case n = 1, we need
to obtain sharp estimates for the p, q-capacitance of conductors ([a, b], (A, B)) with
A < a < b < B. This is the goal of the following proposition.
Proposition 5.2. Suppose n = 1, 1 < p < ∞ and 1 ≤ q ≤ ∞. There exists a constant
C(p, q) ≥ 1 such that
C(p, q)−1 (σ11−p + σ21−p ) ≤ capp,q ([a, b], (A, B)) ≤ C(p, q)(σ11−p + σ21−p ),
where σ1 = a − A and σ2 = B − b.
Proof. By the behaviour of the Lorentz p, q-quasinorm in q (see for instance [BS, Proposition IV.4.2]), it suffices to find the upper bound for the p, 1-capacitance and the lower
bound for the p, ∞-capacitance of the conductor ([a, b], (A, B)). We start with the upper bound for the p, 1-capacitance of this conductor.
We use the function u : (A, B) → R defined by

 1 if a < x < b
x−A
if A < x < a
u(x) =
σ1
 B−x
if b < x < B.
σ2
Then from Lemma 3.1 it follows that u ∈ W1 ([a, b], (A, B)) with

if a < x < b
 0
σ1 −1 if A < x < a
|u0 (x)| =
 σ −1 if b < x < B.
2
We want to compute an upper estimate for ||u0 ||Lp,1 ((A,B),m1 ) . We have
(15)
||u0 ||Lp,1 ((A,B),m1 ) ≤ ||σ1 −1 ||Lp,1 ((A,a),m1 ) + ||σ2 −1 ||Lp,1 ((b,B),m1 )
−1+1/p
−1+1/p
= p σ1
+ σ2
.
Therefore
capp,1 ([a, b], (A, B)) ≤ C(p)(σ11−p + σ21−p ).
We try to get lower estimates for the p, ∞-capacitance of this conductor. Let v ∈
W ([a, b], (A, B)) be an arbitrary admissible function. We let v1 be the restriction of
v to (A, a) and v2 be the restriction of v to (b, B) respectively. We notice that v 0 is
supported in (A, a) ∪ (b, B). Therefore, since v 0 coincides with v10 on (A, a) and with v20
on (b, B), we have that
(16)
||v 0 ||Lp,∞ ((A,B),m1 ) ≥ max(||v10 ||Lp,∞ ((A,a),m1 ) , ||v20 ||Lp,∞ ((b,B),m1 ) ).
From ([Cos, Corollary 2.4]) we have
−1/p0
||v10 ||Lp,∞ ((A,a),m1 ) ≥ 1/p0 · σ1
11
||v10 ||L1 ((A,a),m1 ) .
Since
||v10 ||L1 ((A,a),m1 )
Z
a
|v10 (x)|dx ≥ 1,
=
A
we obtain
−1/p0
||v10 ||Lp,∞ ((A,a),m1 ) ≥ 1/p0 · σ1
(17)
Similarly, since
||v20 ||L1 ((A,a),m1 )
Z
.
a
=
|v20 (x)|dx ≥ 1,
A
we obtain
(18)
−1/p0
||v20 ||Lp,∞ ((b,B),m1 ) ≥ 1/p0 · σ2
.
From (16), (17) and (18) we get the desired lower bound for the p, ∞-capacitance. This
finishes the proof.
Now we state and prove a necessary and sufficient two-weight imbedding condition
for the case n = 1.
Theorem 5.3. Suppose n = 1. Let p, q, r, s be chosen such that 1 < p < ∞, 1 ≤ q < ∞
and 1 < s ≤ max(p, q) ≤ r < ∞. Let Ω be an open set in R and let µ and ν be two
nonnegative locally finite measures on Ω.
(i) Suppose that 1 ≤ q ≤ p. The inequality
||f ||Lr,p (Ω,µ) ≤ A ||f 0 ||Lp,q (Ω,m1 ) + ||f ||Ls,p (Ω,ν)
(19)
holds for every f ∈ Lip0 (Ω) if and only if there exists a constant K > 0 such that the
inequality (7) is valid whenever x, d and τ are such that σd+τ (x) ⊂ Ω.
(ii) Suppose that p < q < ∞. The inequality
(20)
||f ||Lr,q (Ω,µ) ≤ A ||f 0 ||Lp,q (Ω,m1 ) + ||f ||Ls,q (Ω,ν)
holds for every f ∈ Lip0 (Ω) if and only if there exists a constant K > 0 such that the
inequality (7) is valid whenever x, d and τ are such that σd+τ (x) ⊂ Ω.
Proof. We only have to prove that the sufficiency condition for intervals implies the
sufficiency condition for general bounded and open sets g and G with g ⊂ G ⊂ G ⊂ Ω.
Let G be the union of nonoverlapping intervals Gi and let gi = G ∩ gi . We denote
by hi the smallest interval containing gi and by τi the minimal distance from hi to
R \ Gi . We also denote by Hi the open interval concentric with hi such that the
minimal distance from hi to R \ Hi is τi . Then Hi ⊂ Gi . From Remark 3.3 we have
that capp,q (gi , Gi ) = capp,q (hi , Gi ). Moreover, from Theorem 3.2 (ii) and Proposition
5.2 we have
C(p, q)−1 τi1−p ≤ capp,q (hi , Gi ) ≤ capp,q (hi , Hi ) ≤ 2 C(p, q)τi1−p
for some constant C(p, q) ≥ 1. Since g is compact lying in ∪i≥1 Gi , it follows that g is
covered by only finitely many of the sets Gi . This and Theorem 3.2 (ix) allow us to
assume that G is in fact written as a finite union of disjoint intervals Gi . Now the proof
splits, depending on whether 1 ≤ q ≤ p or p < q < ∞.
We assume first that 1 ≤ q ≤ p. We have
X
X
(21)
capp,q (g, G) ≥
capp,q (gi , Gi ) =
capp,q (hi , Gi ).
i
i
12
Using (7), we obtain
µ(gi )p/r ≤ µ(hi )p/r ≤ K1 (τi1−p + ν(Hi )p/s )
≤ K1 C(p, q)(capp,q (gi , Gi ) + ν(Gi )p/s )
where K1 is a positive constant independent of g and G. Since s ≤ p ≤ r < ∞, we
have
X
µ(g)p/r ≤
µ(gi )p/r
i
and
X
ν(Gi )p/s ≤ ν(G)p/s .
i
This and (21) prove the claim when 1 ≤ q ≤ p.
We assume now that p < q < ∞. We have
X
X
capp,q (g, G)q/p ≥
(22)
capp,q (gi , Gi )q/p =
capp,q (hi , Gi )q/p .
i
i
Using (7), we obtain
q(1−p)/p
µ(gi )q/r ≤ µ(hi )q/r ≤ K2 (τi
+ ν(Hi )q/s )
≤ K2 C(p, q)q/p (capp,q (gi , Gi )q/p + ν(Gi )q/s )
where K2 is a positive constant independent of g and G. Since s ≤ q ≤ r < ∞, we have
X
µ(g)q/r ≤
µ(gi )q/r
i
and
X
ν(Gi )q/s ≤ ν(G)q/s .
i
This and (22) prove the claim when p < q < ∞. The theorem is proved.
Acknowledgements. Both authors were partially supported by NSERC and by the
Fields Institute (Ş. Costea) and by NSF grant DMS 0500029 (V. Maz’ya).
References
[Ad]
[AH]
[AP]
[AX1]
[AX2]
[Ai]
[Alb]
[AFT1]
D.R. Adams. On the existence of capacitary strong type estimates in Rn . Ark. Mat. 14
(1976), 125–140.
D.R. Adams, L.I. Hedberg. Function Spaces and Potential Theory, Springer Verlag, 1996.
D.R. Adams, M. Pierre. Capacitary strong type estimates in semilinear problems. Ann.
Inst. Fourier (Grenoble) 41 (1991), 117–135.
D.R. Adams, J. Xiao. Strong type estimates for homogeneous Besov capacities. Math.
Ann. 325 (2003), no.4, 695–709.
D.R. Adams, J. Xiao. Nonlinear potential analysis on Morrey spaces and their capacities.
Indiana Univ. Math. J. 53 (2004), no.6, 1631–1666.
H. Aikawa. Capacity and Hausdorff content of certain enlarged sets. Mem. Fac. Sci. Eng.
Shimane. Univ. Series B: Mathematical Science 30 (1997), 1–21.
A. Alberico. Moser type inequalities for higher-order derivatives in Lorentz spaces. To
appear in Potential Anal.
A. Alvino, V. Ferone, G. Trombetti. Moser-type inequalities in Lorentz spaces. Potential
Anal. 5 (1996), no. 3, 273–299.
13
[AFT2]
A. Alvino, V. Ferone, G. Trombetti. Estimates for the gradient of solutions of nonlinear
elliptic equations with L1 data. Ann. Mat. Pura Appl. 178 (2000), no. 4, 129–142.
[BBGGPV] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vázquez. An L1 -theory
of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm.
Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 241–273.
[BS]
C. Bennett, R. Sharpley. Interpolation of Operators, Academic Press, 1988.
[CS]
Z.-Q. Chen, R. Song. Conditional gauge theorem for non-local Feynman-Kac transforms.
Probab. Theory Related Fields. 125 (2003), no. 1, 45–72.
[CHK]
H.-M. Chung, R.A. Hunt, and D.S. Kurtz. The Hardy-Littlewood maximal function on
L(p, q) spaces with weights. Indiana Univ. Math. J. 31 (1982), no. 1, 109–120.
[Cia]
A. Cianchi. Moser-Trudinger inequalities without boundary conditions and isoperimetric
problems. Indiana Univ. Math. J. 54 (2005), no. 3, 669–705.
[CP]
A. Cianchi and L. Pick. Sobolev embeddings into BMO, VMO, and L∞ . Ark. Mat. 36
(1998), 317–340.
[Cos]
Ş. Costea. Scaling invariant Sobolev-Lorentz capacity on Rn . Indiana Univ. Math. J. 56
(2007), no. 6, 2641–2670.
[DKX]
G. Dafni, G. Karadzhov, J. Xiao. Classes of Carleson-type measures generated by capacities. Math. Z. 258 (2008), no.4, 827–844.
[Dah]
B. Dahlberg. Regularity properties of Riesz potentials. Indiana Univ. Math. J. 28 (1979),
no. 2, 257–268.
[DHM]
G. Dolzmann, N. Hungerbühler, S. Müller. Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side. J. Reine
Angew. Math. 520 (2000), 1–35.
[Fed]
H. Federer. Geometric Measure Theory, Springer-Verlag, 1969.
[Fi]
P.J. Fitzsimmons. Hardy’s inequality for Dirichlet forms. J. Math. Anal. and Appl. 250
(2000), 548–560.
[FU1]
M. Fukushima, T. Uemura. On Sobolev and capacitary inequalities for contractive Besov
spaces over d-sets. Potential Analysis. 18 (2003), no. 1, 59–77.
[FU2]
M. Fukushima, T. Uemura. Capacitary bounds of measures and ultracontractivity of time
changed processes. J. Math. Pures Appl. 82 (2003), no. 5, 553–572.
[Gr]
A. Grigor’yan. Isoperimetric inequalities and capacities on Riemannian manifolds. Operator Theory, Advances and Applications, Vol. 110, The Maz’ya Anniversary Collection,
vol. 1, Birkhäuser, 1999, 139–153.
[Haj]
P. Hajlasz. Sobolev inequalities, truncation method, and John domains. Rep. Univ.
Jyväskylä Dep. Math. Stat., 83, Jyväskylä, 2001.
[Han]
K. Hansson. Embedding theorems of Sobolev type in potential theory, Math. Scand. 45
(1979), 77–102.
[HMV]
K. Hansson, V. Maz’ya, I.E. Verbitsky. Criteria of solvability for multi-dimensional Riccati’s equation. Arkiv för Matem. 37 (1999), no. 1, 87–120.
[HL]
S. Hudson, M. Leckband. A sharp exponential inequality for Lorentz-Sobolev spaces on
bounded domains. Proc. Amer. Math. Soc. 127 (1999), no. 7, 2029–2033.
[Ka]
V. Kaimanovich. Dirichlet norms, capacities and generalized isoperimetric inequalities
for Markov operators. Potential Anal. 1 (1992), no. 1, 61–82.
[KKM]
J. Kauhanen, P. Koskela, J. Malý. On functions with derivatives in a Lorentz space.
Manuscripta Math. 100 (1999), no. 1, 87–101.
[Ko1]
T. Kolsrud. Condenser capacities and removable sets in W 1,p . Ann. Acad. Sci. Fenn. Ser.
A. Math. 8 (1983), 343–348.
[Ko2]
T. Kolsrud. Capacitary integrals in Dirichlet spaces. Math. Scand. 55 (1984), 95–120.
[Mal]
J. Malý. Sufficient conditions for change of variables in integral. Proceedings on Analysis
and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk.
Sib. Otdel. Inst. Mat., Novosibirsk, (2000), 370–386.
14
[M1]
[M2]
[M3]
[M4]
[M5]
[MN]
[MP]
[Ne]
[Ra]
[ST]
[Ta]
[V1]
[V2]
[Vo]
V. G. Maz’ya. On certain integral inequalities for functions of many variables. Problems
of Mathematical Analysis, Leningrad Univ. 3 (1972), 33–68 (Russian). English trans. J.
Soviet Math. 1 (1973), 205–234.
V.G. Maz’ya. Summability with respect to an arbitrary measure of functions from S.
L. Sobolev-L. N. Slobodeckiı̆ spaces. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst.
Steklov. (LOMI) 92 (1979), 192–202.
V.G. Maz’ya. Sobolev Spaces, Springer, 1985.
V. Maz’ya. Conductor and capacitary inequalities for functions on topological spaces and
their applications to Sobolev type imbeddings. J. Funct. Anal. 224 (2005), no.2, 408–430.
V. Maz’ya. Conductor inequalities and criteria for Sobolev type two-weight imbeddings.
J. Comput. Appl. Math. 194 (2006), no. 11, 94–114.
V. Maz’ya, Yu. Netrusov. Some counterexamples for the theory of Sobolev spaces on bad
domains. Potential Anal. 4 (1995), 47–65.
V. Maz’ya, S. Poborchi. Differentiable Functions on Bad Domains. World Scientific, 1997.
Yu.V. Netrusov. Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel
type. Trudy Math., Inst. Steklov 187 (1990), 185–203.
M. Rao. Capacitary inequalities for energy. Israel J. of Math. 61 (1988), no.1, 179–191.
S. Serfaty, I. Tice. Lorentz space estimates for the Ginzburg-Landau energy. J. Funct.
Anal. 254 (2008), no. 3, 773–825.
M. Takeda. Lp -independence of the spectral radius of symmetric Markov semigroups.
Canadian Math. Soc. Conference Proceedings 29 (2000), 613–623.
I.E. Verbitsky. Superlinear equations, potential theory and weighted norm inequalities.
Proceedings of the Spring School VI, Prague, May 31-June 6, 1998.
I.E. Verbitsky. Nonlinear potentials and trace inequalities. Operator Theory, Advances
and Applications, Vol. 110, The Maz’ya Anniversary Collection, vol. 2, Birkhäuser, 1999,
323–343.
Z. Vondraček. An estimate for the L2 -norm of a quasi continuous function with respect
to smooth measure, Arch. Math. 67 (1996), 408–414.
Şerban Costea: [email protected], McMaster University, Department
of Mathematics and Statistics, 1280 Main Street West, Hamilton, Ontario L8S 4K1,
Canada, and The Fields Institute for Research in Mathematical Sciences, 222 College
Street, Second Floor, Toronto, Ontario, M5T 3J1, Canada
Vladimir Maz’ya: [email protected], [email protected], [email protected],
Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA, Department of Mathematical Sciences, M O Building, University
of Liverpool, Liverpool L69 3BX, UK, and Department of Mathematics, Linköping
University, SE-581 83 Linköping, Sweden
15